# An object with a mass of 4 kg is acted on by two forces. The first is F_1= < -7 N , 4 N> and the second is F_2 = < -3 N, -2 N>. What is the object's rate and direction of acceleration?

May 5, 2016

The first force vector acting on the object of mass $m = 4 k g$ is given by ${\vec{F}}_{1} = \left(- 7 \hat{i} + 4 \hat{j}\right) N$

The 2nd force vector acting on the object of mass $m = 4 k g$ is given by ${\vec{F}}_{2} = \left(- 3 \hat{i} - 2 \hat{j}\right) N$
So the resultant force
$\vec{F} = {\vec{F}}_{1} + {\vec{F}}_{2} = \left(\left(- 7 - 3\right) \hat{i} + \left(4 - 2\right) \hat{j}\right) N = \left(- 10 \hat{i} + 2 \hat{j}\right) N$

So the acceleration
$\vec{a} = \frac{\vec{F}}{m} = \frac{1}{4} \left(- 10 \hat{i} + 2 \hat{j}\right) \frac{m}{s} ^ 2$$= \frac{1}{2} \cdot \left(- 5 \hat{i} + \hat{j}\right) \frac{m}{s} ^ 2$

Hence the magnitude of acceleration is
$a = \sqrt{{\left(- \frac{5}{2}\right)}^{2} + {\left(\frac{1}{2}\right)}^{2}} = \frac{\sqrt{26}}{2} \frac{m}{s} ^ 2$

And its direction is ${\tan}^{-} 1 \left(- \frac{1}{5}\right)$ with the X-axis.